Local analytic conjugacy of resonant analytic mappings in two variables, in the non-archimedean setting

Abstract

In this note, we consider locally invertible analytic mappings in two dimensions, with coefficients in a non-archimedean field. Suppose such a map has a Jacobian with eigenvalues $\lambda_1$ and $\lambda_2$ so that $|\lambda_1|>1$ and $\lambda_2$ is a positive power of $\lambda_1$, or that $\lambda_1=1$ and $|\lambda_2|\neq 1$. We prove that two formal maps with eigenvalues satisfying either of these conditions are analytically equivalent if and only if they are formally equivalent.